124 Lesson 4.1 ~ Multiplication Properties of Exponents

mUlTiplicaTion propErTiEs of ExponEnTs

lesson 4.1

When a numerical expression is the product of a repeated factor, it can be written using a power. A power consists of two parts, the base and the exponent. The base of the power is the repeated factor. The exponent shows the number of times the factor is repeated.

It is important to know how to read powers correctly.

Power reading the expression expanded Form Value

52

“five to the second power” or

“five squared”5 × 5 25

63

“six to the third power”or

“six cubed”6 × 6 × 6 216

24 “two to the fourth power” 2 × 2 × 2 × 2 16

Use expanded form to discover two different exponent multiplication properties.

step 1: Write each of the following products in expanded form. a. 5³ ∙ 5⁴ b. 4² ∙ 4⁸ c. x³ ∙ x²

step 2: Rewrite each of the products in step 1 as a single term with one base and one exponent.

step 3: What relationship do you see between the original bases and the single term’s base? What about the original exponents and the single term’s exponent?

step 4: Based on your findings, write a statement explaining how to find the product of two powers with the same base WITHOUT writing the terms in expanded form.

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Lesson 4.1 ~ Multiplication Properties of Exponents 125

step 5: Write each of the following powers in expanded form. Then rewrite the power as a single term. The first one is done for you. a. ( 3² ) ⁴ → ( 3² ) ( 3² ) ( 3² ) ( 3² ) → 3 ∙ 3 ∙ 3 ∙ 3 ∙ 3 ∙ 3 ∙ 3 ∙ 3 → 3⁸

b. ( 7³ ) ²

c. ( x³ ) ⁵

step 6: What is the relationship between the final exponent and the power to a power? Based on your findings, write a statement explaining how to find the power of a power WITHOUT expanding the power.

You can also simplify a power of a product. Look at ( df ) ³.

Written in expanded form: ( df ) ( df ) ( df )

Group like variables together: ( d ∙ d ∙ d ) ( f ∙ f ∙ f )

Simplify: d ³f ³

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126 Lesson 4.1 ~ Multiplication Properties of Exponents

simplify each of the following.

a. y ³x ²y ⁶x b. ( b ³w ² ) ⁴ c. ( 5p⁴ ) ( 2p ³ )

a. Group like variables together. y ³x ²y ⁶x = y ³y ⁶ ∙ x ²x Add exponents with the same base. y ³+⁶x ²+¹ = y⁹x³

b. Distribute the exponent to each base. ( b ³w ² ) ⁴ = ( b ³ ) ⁴ ( w ² ) ⁴ Multiply exponents . ( b ³ ) ⁴ ( w ² ) ⁴ = b ¹²w ⁸

c. Group like terms together. ( 5p ⁴ ) ( 2p ³) = ( 5 ∙ 2 ) ( p ⁴p ³ )

Multiply coefficients. Add ( 5 ∙ 2 ) ( p ⁴p ³) = 10p ⁷ exponents with the same base.

A simplified expression should have: ◆ each base appear exactly once, ◆ no powers to powers, ◆ no numeric values with powers, and ◆ fractions written in simplest form.

simplify each of the following.

a. 6x²y⁴z³ ∙ 3x ⁵z ² b. ( 4m³w ) ² ( 5m²w ² ) ³

a. Rewrite the expression. 6x ²y ⁴z ³ ∙ 3x ⁵z ² Group like terms together. 6 ∙ 3 ∙ x ² ∙ x ⁵ ∙ y⁴ ∙ z ³ ∙ z ²

Multiply coefficients. Add 18x⁷y⁴z ⁵ exponents with the same base.

b. Rewrite the expression. ( 4m³w ) ² ( 5m²w² ) ³

Distribute the exponent to each base. 4² ( m³ ) ²(w)² ∙ 5³ ( m² ) ³ ( w ² ) ³

Find the values of the coefficients 16 ( m³ ) ²(w)² ∙ 125 ( m² ) ³ ( w ² ) ³ with exponents.

Multiply exponents. 16m⁶w ² ∙ 125m⁶w ⁶

Multiply coefficients. Add 2000m¹²w ⁸ exponents with the same base.

ExamplE 1

solutions

ExamplE 2

solutions

Lesson 4.1 ~ Multiplication Properties of Exponents 127

ExErcisEs

simplify. 1. x ³x ² 2. ( y ⁵ ) ² 3. ( pq ) ⁵

4. ( 4x ⁵ ) ³ 5. ( w ²y⁴z ⁶ ) ( w ⁵y ³z ) 6. ( 2a ⁶b ) ( 3a³b³ )

7. ( 5gh² ) ² 8. ( 9x ⁴y⁵ ) ( −2x ²y⁷ ) 9. ( 0.5f ²d ⁹ ) ³

10. A farmer wants to fence in a square pasture for his sheep. The length of one side of the pasture is represented by kp³. What term represents the area of the pasture?

11. Write and solve a problem that requires you to multiply exponents.

12. Write and solve a problem that requires you to add exponents.

13. Jake, Joe and Jenny attempted to rewrite ( 5² ) ( 4³ ) as a single power. Whose answer is correct? Explain your reasoning.

Jake

( 5² ) ( 4³ ) = 20⁵

Joe ( 5² ) ( 4³ ) cannot be written as a single

power.

Jenny

( 5² ) ( 4³ ) = 9⁵

simplify. 14. ( 3x ² ) ³ ( 4x ⁵ ) ² 15. ( −4y ⁵w ² ) ² ( 2y ⁴ ) ³ 16. ( 5p³ ) ( 5p² ) ³

17. ( 2x ² ) ³ ( 3x ⁴ ) ² ( −2x ³ ) ³ 18. ( −2y ² ) ( −3xy ⁴ ) ² ( 5x ⁶ ) ² 19. ( 4a²b ) ³ ( 5a ⁴b ⁵ ) ²

write the volume of each rectangular prism as a single term.

20.

x ³x ³

x ³ 21.

ab ² a

a ³b ⁴

22.

y ³5y ²

3y ²

23. The length of a rectangle is five times its width. If its width is x³ units, what is the area of the rectangle?

24. The width of a rectangular prism is four times its length. The height is three times its length. If its length is m units, what is the volume of the prism? Show all work necessary to justify your answer.

128 Lesson 4.1 ~ Multiplication Properties of Exponents

rEviEw

Graph the pre-image and the image under the given transformation. label the vertices correctly.

25. Triangle S(4, 5), W(2, 4) and R(3, 1) (x, y) → (y, −x)

26. Rectangle N(3, −3), I(−3, −3), C(−3, −1) and E(3, −1) Reflection over the x-axis

27. Triangle P(3, 1), A(6, −2) and L(4, 2) (x, y) → (x – 2, y + 5)

28. Jill moved a figure 3 units down and 6 units right. She predicted that the resulting figure would be congruent to her original shape. Do you agree or disagree? Explain your reasoning.

29. What type of transformation creates a figure similar to its pre-image, but not congruent?

30. A figure is reflected over the x-axis and then reflected over the y-axis. Finally, the figure is dilated with a scale factor of 1.5. Is the image similar or congruent to the original figure? Explain how you know your answer is correct.

31. A triangle has vertices at A(−4, 1), B(2, 1) and C(2, 9). a. Find the perimeter of ∆ABC. Use mathematics to justify your answer. b. Carmen translated the triangle 3 units left and 2 units up. What is the perimeter of ∆A’B’C’? Use words and/or numbers to show how you determined your answer.

tic-tAc-toe ~ the sol A r sy ste m

There are five basic solids: prisms, cylinders, pyramids, cones and spheres. The surface area of a sphere is not the sum of the lateral area and the base area. The formula is given below.

surface Area of a sphereThe surface area of a sphere is the productof 4, pi and the radius squared.

surface Area = 4πr²

1. Use resources to find the radius of the eight planets in our solar system.

2. Assume each planet is a perfect sphere. Calculate the surface area of each planet in square miles. Use 3.14 for π.

3. Create a booklet showing the eight planets and their surface areas. Include at least one other interesting fact about each planet.